Problem: Talulah is an ecologist who studies the change in the penguin population of Antarctica over time. She observed that the population decays by a factor of $\dfrac{8}{9}$ every $4$ months. The population of penguins can be modeled by a function, $P$, which depends on the amount of time, $t$ (in months). When Talulah began the study, she observed that there were $27{,}000$ penguins in Antarctica. Write a function that models the population of the penguins $t$ months since the beginning of Talulah's study. $P(t) = $
Answer: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of penguins is $27{,}000$, and the population decays by a factor of $\dfrac 89$ in $4$ months. This means that the initial quantity is $A=27{,}000$ and the factor is $B=\dfrac{8}{9}$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac{8}{9}$ every $4$ months. Finding the expression in the exponent We know that the population of penguins is multiplied by $\dfrac{8}{9}$ every $4$ months. This means that each time $t$ increases by $4$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{4}$. When the initial measurement is made, the population hasn't changed. So $P(0) = 27{,}000$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{4}$. Summary We found that the following function models the population of penguins $t$ months since the beginning of Talulah's study. P ( t ) = 27,000 ⋅ ( 8 9 ) t 4 P(t)=27{,}000\cdot \left(\dfrac{8}{9}\right)\^{ \frac{t}{4}}